Agriculture Reference
In-Depth Information
is the cost of measuring and dividing each input. Without considering these
measurement costs,
where
m
s
∗
and
q
∗
satisfy
5
e
s
(h
e
−
w)
+
l
s
(h
l
−
r)
+
k
s
(h
k
−
c)
=
0,
and
(5.3)
e
q
(h
e
−
w)
+
l
q
(h
l
−
r)
+
k
q
(h
k
−
c)
=
0.
(5.4)
If input measurement costs are ignored, the solutions to equations (5.3) and (5.4) de-
termine the optimal sharing rules,
q
∗
. If, however, there are costs of measuring and
dividing the input, they must be considered in order for
s
∗
and
s
∗
and
q
∗
to be derived. If these costs
(m)
are lump-sum expenditures and the inputs are unrelated, then the solution to equations
(5.3) and (5.4) is dichotomous and the choice of the optimal cropshare is made after the
decision about input cost sharing is made.
6
As long as these measurement costs are worth incurring, the model implies that input-
sharing and output-sharing rules will be identical; that is,
q
∗
=
s
∗
. If these costs are not
worth incurring, however, then input costs will not be shared and the farmer will bear all
costs; that is,
s
∗
<q
∗
=
1. This is proved by examining (5.4) and considering the case for
which measurement costs are small and worth incurring. Because inputs are independent,
e
q
=
l
q
=
0. Also, because
k
q
<
0, a drop in the farmer's cost of input
k
leads him to choose
an increased amount of the input. Thus, equation (5.4) implies that
h
k
=
c
. The first-order
sh
k
(k
s
)
≡
qc
q
∗
=
s
∗
.
condition in the farmer's problem,
, must also hold, implying that
Therefore, when inputs are shared the optimal sharing rules must satisfy
e
s
(h
e
−
w)
+
l
s
(h
l
−
r)
=
0,
and
(5.5)
q
∗
=
s
∗
.
(5.6)
The optimal use of inputs is illustrated by figure 5.1, which considers the case where
input shares equal output shares.
7
For clarity the marginal product curves for each input are
again drawn as identical and linear. Each panel shows the marginal product curve for each
input; it also shows the farmer's share of the marginal product that depends on the optimal
share
s
∗
. Because the farmer shares the crop, he applies less capital and labor, which results
in a marginal distortion equal to AB. The landowner's inability to fully price the land's
attributes results in overuse by the farmer and a marginal distortion equal to CD. For the
other variable inputs, sharing rules yield no marginal distortion, because input and output
shares are identical. For the value of the contract to be maximized, the optimal output share
s
∗
must be chosen so that the marginal distortions on inputs
e
and
l
exactly offset (AB=
CD).
8
When measurement costs are large and not worth incurring, the farmer must bear all
of the input costs; that is,
q
=
1. The assumption of independent inputs still implies that
e
q
=
l
q
=
0; but now
h
k
=
c
, because
s
=
q
and
s<
1 in any cropshare contract, which also
